Several fundamental results have been obtained about hypersurfaces with sufficiently small entropy, starting with work of Colding–Ilmanen–Minicozzi–White [1]} who proved that the round sphere \(\mathbb {S}^n(\sqrt{2n})\)  has minimal entropy among all closed self-shrinkers. This was extended by Bernstein–Wang [2]} who showed that the round sphere minimizes entropy among all closed hypersurfaces (see also [3]}, [4]}). Moreover, Bernstein–Wang have also proven [5]} that the cylinder \(\mathbb {S}^1(\sqrt{2})\times \mathbb {R}\subset \mathbb {R}^3\)  has second least entropy among all self-shrinkers in \(\mathbb {R}^3\)  (their result crucially relies on Brendle's classification of genus zero self-shrinkers [6]}).
